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Theorem resieq 4565
Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
resieq ((B A 𝐶 A) → (B( I ↾ A)𝐶B = 𝐶))

Proof of Theorem resieq
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 breq2 3759 . . . . 5 (x = 𝐶 → (B( I ↾ A)xB( I ↾ A)𝐶))
2 eqeq2 2046 . . . . 5 (x = 𝐶 → (B = xB = 𝐶))
31, 2bibi12d 224 . . . 4 (x = 𝐶 → ((B( I ↾ A)xB = x) ↔ (B( I ↾ A)𝐶B = 𝐶)))
43imbi2d 219 . . 3 (x = 𝐶 → ((B A → (B( I ↾ A)xB = x)) ↔ (B A → (B( I ↾ A)𝐶B = 𝐶))))
5 vex 2554 . . . . 5 x V
65opres 4564 . . . 4 (B A → (⟨B, x ( I ↾ A) ↔ ⟨B, x I ))
7 df-br 3756 . . . 4 (B( I ↾ A)x ↔ ⟨B, x ( I ↾ A))
85ideq 4431 . . . . 5 (B I xB = x)
9 df-br 3756 . . . . 5 (B I x ↔ ⟨B, x I )
108, 9bitr3i 175 . . . 4 (B = x ↔ ⟨B, x I )
116, 7, 103bitr4g 212 . . 3 (B A → (B( I ↾ A)xB = x))
124, 11vtoclg 2607 . 2 (𝐶 A → (B A → (B( I ↾ A)𝐶B = 𝐶)))
1312impcom 116 1 ((B A 𝐶 A) → (B( I ↾ A)𝐶B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755   I cid 4016  cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  foeqcnvco  5373  f1eqcocnv  5374
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